Copyright © Philip M. Parker, INSEAD. Terms of Use.

Curvature

Definition: Curvature

Curvature

Noun

1. (medical) a curving or bending; often abnormal; "curvature of the spine".

2. The rate of change (at a point) of the angle between a curve and a tangent to the curve.

3. The property possessed by the curving of a line or surface.

Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved.
 

Date "curvature" was first used in popular English literature: sometime before 1781. (references)


Synonym: Curvature

Synonym: curve (n). (additional references)

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Specialty Definition: Curvature

(From Wikipedia, the free Encyclopedia)

Curvature is the amount by which a curve, surface, or other manifold deviates from a straight line or (hyper)plane.

Curvature of curves

For a plane curve C, the curvature at a given point P has magnitude equal to the reciprocal of the radius of an osculating circle (a circle that "kisses" or closely touches the curve at the given point), and is a vector pointing in the direction of that circle's center. The magnitude of curvature at points on physical curves can be measured in diopterss (alternative spelling: dioptre); a diopter is one per meter.

The smaller the radius r of the osculating circle, the larger the magnitude of the curvature (1/r) will be; so that where a curve is "nearly" straight, the curvature will be close to zero, and where the curve undergoes a tight turn, the curvature will be large in magnitude.

A straight line has everywhere curvature 0; a circle of radius r has everywhere curvature of magnitude 1/r.

Local expressions

For a plane curve given parametrically as the curvature is

where the dots denote differentiation respect to t.

For a plane curve given implicitly as the curvature is

that is, the divergence of the direction of the gradient of f. This last formula also gives the mean curvature of an hypersurface in euclidean space.

Curvature of surfaces

For two-dimensional surfaces embedded in R3, there are two kinds of curvature: Gaussian (or scalar) curvature, and Mean curvature. To compute these at a given point of the surface, consider the intersection of the surface with a plane containing a fixed normal vector at the point. This intersection is a plane curve and has a curvature; if we vary the plane, this curvature will change, and there are two extremal values - the maximal and the minimal curvature, called the main curvatures, 1/R1 and 1/R2. Here we adopt the convention that a curvature is taken to be positive if its vector points in the same direction as the surface's chosen normal, otherwise negative.

The Gaussian curvature is equal to the product 1/R1R2. It has the dimension of 1/length2 and is everywhere positive for spheres, everywhere negative for hyperboloids and everywhere zero for planes. It determines whether a surface has elliptic (when it is positive) or hyperbolic (when it is negative) geometry at a point.

The above definition of Gaussian curvature is extrinsic in that it uses the surface's embedding in R3, normal vectors, external planes etc. Gaussian curvature is however in fact an intrinsic property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. Formally, Gaussian curvature only depends on the surface's structure as a Riemannian manifold. This is Gauss' celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking.

An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r. She runs around P while the thread is completely stretched and measures the length C(r) of one complete trip around P. If the surface were flat, she would find C(r) = 2πr. On curved surfaces, the formula for C(r) will be different, and the Gaussian curvature K at the point P can be computed as

The integral of the Gaussian curvature over the whole surface is closely related to the surface's Euler characteristic; see the Gauss-Bonnet theorem.

The Mean curvature is equal to the sum of the main curvatures 1/R1+1/R2. It has the dimension of 1/length. A minimal surface like a soap film or soap bubble has mean curvature zero. The mean curvature depends on the embedding and is not an intrinstic property of a surface - for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.

Higher dimensions

In the case of higher-dimensional manifolds curvature is defined as a tensor, which depends on a connection. A connection gives a way to transport vectorss (and therefore also tensors) parallelly along a given path on a manifold. Given a metric (or first fundamental form) on a manifold, there is a unique connection which preserves this metric, the Levi Civita connection, and a corresponding curvature tensor.

The curvature tensor tells you what happens if you transport a vector around a small loop. If a loop is approximated by a small parallelogram spanned by two tangent vectors, then transporting a vector around this loop results in a linear transformation of this vector - for each pair of vectors defining a parallelogram, there is a matrix which tells you what change in a tangent space results from the parallel transport along this parallelogram. Thus, curvature is a tensor of type (1,3).

The curvature tensor has the special property that it is antisymmetric in the indices giving a loop (if you reverse your loop you will get the inverse transformation) and is thus a matrix of 2-formss.

The sectional curvature, which depends on the plane of the section, determines curvature tensor completely, and is a good way to think of curvature.

Curvature is intimately related to the holonomy group which is the group of all linear transformations of the tangent space at a point which can result from a parallel transport around a loop. The Bianchi identities restrict the possibilities for these groups, and with the exception of symmetric spaces there are few possibilities given by the Berger list.

Contraction of a full curvature tensor gives the two-valent Ricci-curvature and the scalar curvature. The Ricci-curvature can be used to define Chern classes of a manifold, which are topological invariants independent of the metric. The Einstein equations of general relativity are given in terms of scalar and Ricci curvatures.

See also:

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Curvature."

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Synonyms within Context: Curvature

ContextSynonyms within Context (source: adapted from Roget's Thesaurus).

Curvature

Curvature, curvity, curvation; incurvature, incurvity; incurvation; bend; flexure, flexion, flection; conflexure; crook, hook, bought, bending; deflection, deflexion; inflection, inflexion; concameration; arcuation, devexity, turn, deviation, detour, sweep; curl, curling; bough; recurvity, recurvation; sinuosity.

Source: adapted from Roget's Thesaurus.

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Crosswords: Curvature

English words defined with "curvature": Absolute curvature, Aduncity, Angle of curvature, Angle of torsion, Apophyge, arteria gastrica breves, arteria gastrica sinistra, ateria gastrica dextra ♦ bandy legs, bow leg, bow legs ♦ center of curvature, centre of curvature, Chord of curvature, Chordee, circle of curvature, crookback ♦ Dead rise, Diminishing rule ♦ Equating for curves ♦ gastroepiploic vein, gastroomental vein ♦ hollow-back, humpback, hunchback ♦ Incurvation, Indicatrix ♦ left gastric artery, lordosis, lyrate, lyrate leaf ♦ Ophthalmometer, optic axis, osculating circle, Osculating circle of a curve, Osculatory plane, Osculatory sphere ♦ Point of contrary flexure, Pott's disease, principal axis, Pseudosphere ♦ radius of curvature, right gastric artery ♦ Secondary axis, short gastric artery, Skew curve, spherometer, spinal curvature, straight line ♦ Twisted curve ♦ vasa brevis, vena gastroomentalis. (references)
Specialty definitions using "curvature": assembler-erector ♦ backside grinder, Ballooning Mode, bending-roll operator, bilge plank, bilge plate, bilge trake, Blaton formula ♦ cannage, CHIMNEY SUPERVISOR, BRICK, chord-line, clocker, conformal gears, contact logging device, contact-lens cutter, Corneal Topography, curvature of gravity ♦ diffraction propagation, discontinuity stresses ♦ ELECTRIC-MOTOR-AND-GENERATOR ASSEMBLER ♦ fault plane, FLANGING-ROLL OPERATOR, flat hammerer ♦ GEAR INSPECTOR, geodetic datum, geographical range, glass curvature gauger, GRINDER, HAND ♦ HAMMERSMITH, HANDS ASSEMBLER, hole deviation ♦ increased lordosis ♦ LATHE OPERATOR, CONTACT LENS, law of constant angular momentum, lens assorter, lens grinding, LENS-BLANK GAUGER, LEVEL-VIAL CURVATURE GAUGER ♦ motor-and-generator assembler, MOUTHPIECE MAKER ♦ NAME-PLATE STAMPER ♦ orocline, Orthophotography ♦ PIPE FITTER, DIESEL ENGINE I, PLATE SETTER, FLEXOGRAPHIC PRESS ♦ radio duct, RESTORER, CERAMIC, roll former, ROLL OPERATOR I, rounder and backer, ROUNDING-AND-BACKING-MACHINE OPERATOR ♦ shell sorter, SHELL-GRADER, skeleton tubbing, SPRING INSPECTOR I, standard propagation, stress meter, superstandard propagation, surface wave ♦ tangent distance, tear drop, teariness, theorem of areas, tip chord, transverse curl, two-control-point method, type D region, TYPE-ROLLING-MACHINE OPERATOR ♦ Vacquier-Steenland method ♦ wing chord. (references)

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Commercial Usage: Curvature

DomainTitle

Books

  • Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems: Harmonic Maps and Integrable Systems (reference)

  • Curvature and Betti Numbers (Annals of Mathematics Studies, Number 32) (reference)

  • Curvature and Homology (reference)

  • Lecture Notes on Symmetries and Curvature Structure in General Relativity (Lecture Notes in Physics, Vol 46) (reference)

  • Lectures on Spaces of Nonpositive Curvature (reference)

    (more book examples)

Source: compiled by the editor from various references; see credits.

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Image Slideshow: Curvature

Illustrations:
Curvature

More pictures...

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Photo Album: Curvature

ThumbnailDescription & CreditThumbnailDescription & Credit

"Velocity Field for Fluid Flow Thru a Tube" by Tom Tredon. Use DPGraph's Scrollbar to vary A (the curvature), B (the inner diameter), or C (the speed of the fluid flow). Inspired by a drawing on page 3 of Harry Schey's "Div, Grad, Curl, and All That", 2nd edition. Click on Edit inside DPGraph for more info.

Figure 37. Bucknill-Casella manometric sounder, invented by Lieutenant John T. Bucknill of the Royal Engineers of the Royal Navy in 1870 to mitigate problems with existing sounding systems. This sounder was based on Bourdon's tube, whose curvature varied as a function of the pressure difference between the interior and the fluid in which it was immersed. Louis P. Casella made the final product. Credit: Sailing for Science - the NOAA Fleet Then and Now.

[A tree bound to a stake in an effort to correct the curvature of the trunk]. Credit: National Library of Medicine.

  

Source: pictures compiled by the editor from various references; see picture credits.

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Non-Fiction Usage: Curvature

SubjectTopicQuote

Health

They may alter the cornea's normal curvature, causing periodic blurred vision. (references)

A study that investigated the long-term effects of common approaches to correcting spinal curvature is reviewed. (references)

Epithelial swelling damages vision by changing the cornea's normal curvature, and causing a sight-impairing haze to appear in the tissue. (references)

Source: compiled by the editor from ICON Group International, Inc.; see credits.

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Usage Frequency: Curvature

"Curvature" is generally used as a noun (singular) -- approximately 99.24% of the time. "Curvature" is used about 394 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted)
Parts of SpeechPercentUsage per
100 Million Words		Rank in English
Noun (singular)99.24%39114,189
Lexical Verb (base form)0.51%2245,945
Lexical Verb (infinitive)0.25%1339,140
                    Total100.00%394N/A

Source: compiled by the editor from several corpora; see credits.

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Expressions: Curvature

Expressions using "curvature": Aberrancy of curvature ♦ Absolute curvature ♦ Angle of curvature ♦ center of curvature ♦ Center of curvature of a curve ♦ centre of curvature ♦ Chord of curvature ♦ Circle of curvature ♦ curvature of roadway ♦ curvature of the spine ♦ indicated course curvature ♦ indicated glide path curvature ♦ radius of curvature ♦ spinal curvature. Additional references.

Source: compiled by the editor from various references; see credits.

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Frequency of Internet Keywords: Curvature

The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com.
 
ExpressionFrequency
per Day		ExpressionFrequency
per Day

curvature of the penis

61

curvature scoliosis spine

3

curvature of the spine

46

curvature space time

2

penile curvature

16

c3 c5 cervical curvature

2

curvature radius

11

abnormal curvature penis

2

curvature

10

curvature point

2

curvature earth

9

curvature penis picture

2

spinal curvature

7

curvature point surveying

2

curvature neck

5

correction curvature penis

2

congenital penile curvature

4

can curvature earth see

2

curvature gaussian

3

breathing c3 c5 cervical curvature

2

abnormal curvature spinal

3

curvature lordotic

2

back curvature

3

curvature lordotic

2
Source: compiled by the editor from various references; see credits.

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Modern Translation: Curvature

Language Translations for "curvature"; alternative meanings/domain in parentheses.

Albanian

  

përkulje (bend, Bob, bow, crouch, curtsey, curtsy, droop, flection, flex, flexion, flexure, incurvation, incurvature, inflection, reverence, sag, stoop, warp), lakim (declension, declination, flexure, incurvation, inflection, inflexion), grafik (bend, chart, curve, diagram, graph, graphic, graphical). (various references)

   

Arabic 

  

‏تقويس (bending), ‏إنحناء (bend, bowing, crook, crouch, dipping, drooping, duck, stoop, tendency), ‏إعوجاج, ‏شىء منحن, ‏درجة الإنحناء. (various references)

   

Bulgarian 

  

кривина (camber, incurvation), изкривяване (collapse, contortion, curve, deformation, sag, twist). (various references)

   

Chinese 

  

曲度. (various references)

   

Czech

  

zakřivení (bend). (various references)

   

Danish

  

runding (bending, bilge, camber, curl, fillet, rounding, roundness), krumning (buckling, curve, local buckling, road curve, sweep, warping). (various references)

   

Dutch

  

welving (camber, convexity of roadway, crown, crown of roadway, curvature of roadway, dishing of roadway, plantar arch, swell), kromming (arch, bandelette, brace, camber, crook, curve, road curve, spring, springing, sweep). (various references)

   

Farsi 

  

پیچش (Deflexion, Torsion, Wrest), مقدارانحناء , کجی (Clubfoot, Crook, Hade, List, Obliquity, Skew, Slant, Slope, Tilt), خمیدگی (Bend, Bent, Loop, Offset, Rake, Stoop), انحناء (Deflexion, Wind). (various references)

   

Finnish

  

kaarevuus (arch, arched form, curve). (various references)

   

French

  

courbure (curve). (various references)

   

German

  

krümmung (bend, bending, crook, curve, figure, flexion, meander, turn). (various references)

   

Greek 

  

καμπυλότητα (camber, tortuousity, tortuousness). (various references)

   

Hebrew 

  

עקמומיות (crookedness), עקימות (crookedness), עקום (bending, contortion, crooked, crooking, distortion, twist, warp, wry), עקם. (various references)

   

Hungarian

  

görbület (bend, bending, camber, crook, curve, elbow, flex, long bend, ply), kanyar (bend, corner, curve, loop, turning, twist, wimple). (various references)

   

Indonesian

  

lekukan (dint, indentation). (various references)

   

Italian

  

curvatura (bend, bending, buckle, camber, crook, crookedness, curve, curving, flexion, stoop, warp). (various references)

   

Japanese Kanji 

  

極度 (extreme, maximum, utmost), 曲率 , 曲がり (bend, warp), 曲度 , 反り (arch, curve, warp). (various references)

   

Japanese Katakana 

  

そり (arch, curve, sled, sleigh, warp), きょくど (extreme, maximum, utmost), きょくりつ, まがり (bend, renting a room, warp). (various references)

   

Korean 

  

곡율. (various references)

   

Manx

  

clink (jingle, trick, twist, twist running), cammid (crookedness, obliqueness, wryness). (various references)

   

Pig Latin

  

urvaturecay

   

Portuguese

  

curvatura (bent, buckle, flexure, hog, wind), raio de curvatura (radius). (various references)

   

Romanian

  

curburã (bend, bent, buckle, camber, crack, crook, curve, flexion, flexure, sweep), curbare (flexion, inflexion), curbã (curve, turn, twist, wind), cot (bend, crack, crook, cubit, curve, elbow, Ell, knee, knuckle, loop, meander, sweep, turn, turning, twist, winding), viraj (turn). (various references)

   

Russian 

  

кривизна (curve, flection, flexion, flexure, incurvation, twist, wryness), искривление (contortion, deformation, distortion, hog, twist). (various references)

   

Serbo-Croatian

  

zavoj (bandage, dressing, swathe), kurvatura, krivina (bend, camber, curve, flexure, sinuosity, sinus). (various references)

   

Spanish

  

curvatura (bend, bent). (various references)

   

Swedish

  

krökning (curl, curve, flexion, incurvation, incurvature, loop). (various references)

   

Turkish

  

kavislenme, eğrilik (camber, cast, crookedness, deviousness, obliqueness, skew, slant, tortuosity, unevenness, warp, wryness), eğilme (bending, buckle, dip, droop, flexure, hunch, inclination, inflection, inflexion, lean, proneness, spring, stoop, tilt, tip). (various references)

   

Turkmen 

  

egrilik. (various references)

   

Ukrainian

  

кривизна (anfractuosity, bending, bent, bulging, camber, curve, flection, flexion, hogging), викривлення (contortion, deformation, flexure, warp), згинання (applying, bend, flexure, inflection, inflexion, wriggle). (various references)

   

Vietnamese 

  

sự uốn cong (crook), sự bị uốn cong. (various references)

   

Welsh

  

camedd (bend). (various references)

Source: compiled by the editor from various translation references.

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Derivations & Misspellings: Curvature

Derivations

Words beginning with "curvature": curvatures. (additional references)

Words ending with "curvature": incurvature. (additional references)

Words containing "curvature": incurvatures. (additional references)


Misspellings

"Curvature" is suggested in spellcheckers for the following: Couratari, curvata. (additional references)

Source: compiled by the editor, based on several corpora (additional references).

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Rhyming with "Curvature"

# of Phoneme MatchesPronunciationWord(s) rhyming with "curvature" (pronounced ker"vukher)
3-u kh ercaricature, expenditure, forfeiture, investiture, literature, musculature, signature, tablature, temperature.

Source: compiled by the editor (additional references); see credits.

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Anagrams: Curvature

Scrabble® Enable2K-Verified Anagrams

Words within the letters "a-c-e-r-r-t-u-u-v"

-2 letters: verruca.

-3 letters: acuter, auteur, carter, carver, crater, craver, curare, curate, curter, curvet, tracer.

-4 letters: acute, avert, carer, caret, carte, carve, cater, caver, crate, crave, cruet, crura, curer, curet, curve, cuter, eruct, racer, rater, raver, react, recta, recur, recut, tarre, terra, trace, trave, truce, truer, urare, urate, vertu.

-5 letters: acre, aver, care, carr, cart, cate.

 Words containing the letters "a-c-e-r-r-t-u-u-v"
 

+1 letter: curvatures.

 

+2 letters: incurvature.

 

+3 letters: incurvatures.

Source: compiled by the editor from various references; see credits.

SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.

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INDEX

1. Definition
2. Synonyms
3. Crosswords
4. Usage: Commercial 		5. Images: Slideshow
6. Images: Photo Album
7. Quotations: Non-fiction
8. Usage Frequency 		9. Expressions
10. Expressions: Internet
11. Translations: Modern
12. Derivations 		13. Rhymes
14. Anagrams
15. Bibliography


  

Copyright © Philip M. Parker, INSEAD. Terms of Use.